An unlikely alliance of liberal intellectuals, big donors and Republican strategists has hit on a solution to the influence of big money in politics — even more money.

Starting Monday, the recently formed Mayday super-PAC began a $12 million advertising campaign to help elect lawmakers of both parties who support proposals to diminish the influence of big donors. Founded by a Harvard legal scholar and a former adviser to President George W. Bush, the PAC is the most ambitious effort yet to turn dismay over supersize contributions into a winning political issue.

“Inside-the-Beltway people don’t think this issue matters, they don’t think voters vote on the basis of this issue, and they advise their politicians not to talk about it,” said Lawrence Lessig, a professor at Harvard Law School with ties to Silicon Valley who co-founded the Mayday PAC with Mark McKinnon, a former adviser to Bush. “We think this issue does matter, and we want to prove it.”

As for the contrast between the group’s message and financing, Lessig has adopted a slogan: “Embrace the irony.”

A major part of the effort, which will first take shape in Iowa and New Hampshire, is to fight money with money. The Mayday PAC will begin buying up to a total of $4 million in advertising for both states, where voters historically are receptive to the anti-big-money message, advertising is relatively cheap, and any victories are likely to be noticed by both parties’ future presidential aspirants.

In New Hampshire, the group will seek to help Jim Rubens, a former Republican state senator, defeat Scott Brown, a former U.S. senator from Massachusetts, in the Republican primary for U.S. Senate. In Iowa, the group is backing Staci Appel, a Democrat, for an open congressional seat.

Officials said the PAC is expected to announce its next three targets in August.

The ultimate goal is to reverse a series of defeats that advocates of tighter restrictions on campaign money have suffered since the Supreme Court’s 2010 Citizens United decision. That decision freed corporations and unions to engage in unlimited independent political spending on behalf of candidates.

In the past four years, court rulings and regulatory decisions have only expanded the influence and reach of big donors at the federal and state level.

Many Democratic lawmakers and reform groups have pushed to repeal the decision through a constitutional amendment — the Senate will vote on such a measure later this year — or with legislation that would impose more restrictions on fundraising.

But those efforts have little support among Republicans and face an uphill climb in Congress.

Mayday and allied groups are taking a different tack. Instead of pushing for new restrictions on contributions, they are advocating proposals to spur more giving by small donors, in the hope of diluting the influence of big ones.

“As long as campaign finance is seen as a liberals-only issue, I don’t think we’ll be able to bring in the resources that we need to win,” said Nick Penniman, the executive director of Fund for the Republic. The goup has received money from members of the Democracy Alliance, a club of top liberal donors.

“If you’re a Republican or an independent in this country and you think that money plays too much of a role in politics, you really have no home.”

Mayday’s founders believe that rank-and-file Republicans are more open to limiting the effect of big contributions than their leaders are. McKinnon worked with Juleanna Glover, a Republican lobbyist and former George W. Bush aide, and Trevor Potter, a Republican lawyer who founded the Campaign Legal Center, to interview 60 Republican and conservative leaders, strategists and activists for their views on money and politics.

They found that Obama’s ability to outraise Republican presidential candidates in two straight elections had cut against a common belief in Republican circles that unlimited campaign fundraising and spending gave their party an advantage.

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